Managers of assets, such as portfolios of stocks, projects in a firm, or other assets, typically seek to maximize the expected or average return on an overall investment of funds for a given level of risk as defined in terms of variance of return, either historically or as adjusted using techniques known to persons skilled in portfolio management. Alternatively, investment goals may be directed toward residual return with respect to a benchmark as a function of residual return variance. Consequently, the terms “return” and “variance,” as used in this description and in any appended claims, may encompass, equally, the residual components as understood in the art. The capital asset pricing model of Sharpe and Lintner and the arbitrage pricing theory of Ross are examples of asset evaluation theories used in computing residual returns in the field of equity pricing. Alternatively, the goal of a portfolio management strategy may be cast as the minimization of risk for a given level of expected return.
The risk assigned to a portfolio is typically expressed in terms of its variance σp2 stated in terms of the weighted variances of the individual assets, as:             σ      p      2        =                  ∑        i            ⁢                           ⁢                        ∑          j                ⁢                              w            i                    ⁢                      w            j                    ⁢                      σ                          i              ⁢                                                           ⁢              j                                            ,where wi is the relative weight of the i-th asset within the portfolio, σij=σiσjρij is th covariance of the i-th and j-th assets, ρij is their correlation, and σi is the standard deviation of the i-th asset. The portfolio standard deviation is the square root of the variance of the portfolio.
Following the classical paradigm due to Markowitz, a portfolio may be optimized, with the goal of deriving the peak average return for a given level of risk and any specified set of constraints, in order to derive a so-called “mean-variance (MV) efficient” portfolio using known techniques of linear or quadratic programming as appropriate. Techniques for incorporating multiperiod investment horizons are also known in the art. As shown in FIG. 1A, the expected return μ for a portfolio may be plotted versus the portfolio standard deviation σ, with the locus of MV efficient portfolios as a function of portfolio standard deviation referred to as the “MV efficient frontier,” and designated by the numeral 10. Mathematical algorithms for deriving the MV efficient frontier are known in the art.
Referring to FIG. 1B, a variation of classical Markowitz MV efficiency often used is benchmark optimization. In this case, the expected residual return α relative to a specified benchmark is considered as a function of residual return variance ω, defined as was the portfolio standard deviation σ but with respect to a residual risk. An investor with portfolio A desires to optimize expected residual return at the same level ωA of residual risk. As before, an efficient frontier 10 is defined as the locus of all portfolios having a maximum expected residual return α of each of all possible levels of portfolio residual risk.
Known deficiencies of MV optimization as a practical tool for investment management include the instability and ambiguity of solutions. It is known that MV optimization may give rise to solutions which are both unstable with respect to small changes (within the uncertainties of the input parameters) and often non-intuitive and thus of little investment sense or value for investment purposes and with poor out-of-sample average performance. These deficiencies are known to arise due to the propensity of MV optimization as “estimation-error maximizers,” as discussed in R. Michaud, “The Markowitz Optimization Enigma: Is Optimized Optimal?” Financial Analysts Journal (1989), which is herein incorporated by reference. In particular, MV optimization tends to overweight those assets having large statistical estimation errors associated with large estimated returns, small variances, and negative correlations, often resulting in poor ex-post performance.
Resampling of a plurality of simulations of input data statistically consistent with an expected return and expected standard deviation of return has been applied (see, for example, Broadie, “Computing efficient frontiers using estimated parameters”, 45 Annals of Operations Research 21-58 (1993)) in efforts to overcome some of the statistical deficiencies inherent in use of sample moments alone. Comprehensive techniques based on a resample efficient frontier are described in U.S. Pat. No. 6,003,018 (Michaud et al. '018), issued Dec. 14, 1999, and in the book, R. Michaud, Efficient Asset Management, (Harvard Business School Press, 1998), that MV optimization is a statistical procedure, based on estimated returns subject to a statistical variance, and that, consequently, the MV efficient frontier, as defined above, is itself characterized by a variance. The Michaud patent and book are incorporated herein by reference, as are all references cited in the text of the book.
As taught in the Michaud '018 patent, an MV efficient frontier is first calculated by using standard techniques as discussed above. Since the input data are of a statistical nature (i.e., characterized by means with associated variances and other statistical measures), the input data may be resampled, by simulation of optimization input parameters in a manner statistically consistent with the first set of data, as described, for example, by J. Jobson and B. Korkie, “Estimation for Markowitz Efficient Portfolios,” Journal of Portfolio Management, (1981), which is herein incorporated by reference. Embodiments of the present invention are related to improvements and extensions of the teaching of the Michaud '018 patent.
When portfolios are rebalanced in accordance with current practice, criteria are applied that are typically not portfolio-based or consistent with principles of modern statistics but are generally associated with various ad hoc rules. U.S. Pat. No. 6,003,018 teaches a portfolio-based rebalancing criterion that can be used for all portfolios on the resampled efficient frontier and that is consistent with principles of modern statistics and considers the uncertainty in investment information. However, this rule has some important limitations. Its discriminatory power is not customizable and has too high power at low levels of risk and too little power at high levels of risk. Methods are clearly necessary for providing relatively uniform discriminatory power across portfolio risk levels as well as being able to customize discriminatory power according to the investment needs of organizations which differ in terms of user sophistication, asset class characteristics, or investment strategy requirements. Methods are also clearly desirable that help identify anomalously weighted assets (overly large or small weights) relative to a normal range that is associated with the uncertainty of investment information.